Ed Hellen's Web Page
Dr. Edward Hellen

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Research: Nonlinear dynamics, chaos, excitable systems, Synthetic Genetic Networks, Padé-Laplace, circuits, Lorenz Attractor

Other Stuff: python, electronics, SPICE, publications, racquets, x-country ski in Greensboro

Interesting Dynamics (aka Nonlinear Dynamics )

Chaotic systems:  Real-time bifurcation diagrams produced from analog electronic circuits.  Simple quadratic equations used in finite-difference recursion relations display a wide variety of behaviors including chaos.  Click for article (pdf) or Edward H. Hellen, Am. J. Phys. 72, 499(2004).

Logistic Equation  logistic

Hénon Equation (1-dim)Henon

Control of Chaos: Deterministic chaos (as produced by the above examples) can be controlled by small perturbations of a system parameter (R or a). We derive the transient response for chaos control based on the first difference of the system variable, (xn − xn-1), and verify it experimentally. Graph shows measurements of system values and control parameter taken from controlled Hénon electronic circuit. Here is an arXiv Eprint: J.K. Thomas and E.H. Hellen

Close-up of bifurcation data recorded from Keith's Hénon circuit. Shows period doubling from 4 to 8, period-3, period-5, and period-6 regions.
chaos control

In addition to stabilizing a single fixed point, it is also possible to stabilize unstable periodic orbits. Here we show numerical simulation of control of Period-2 oscillation. System values are red. Perturbations to control parameter are green. Control was turned on at 50, off at 200, back on at 250.
chaos control

Excitable Systems: Fitzhugh-Nagumo. Here are a couple of VPython avi movies. Fitzhugh-Nagumo1 (shows phase-space), Fitzhugh-Nagumo2 (shows phase-space and time evolution). The system includes both a steady leakage current and a pulsed stimulation. Here is a spatiotemporal version showing propagation of excitations too close together: Fitz-Nag_Diffusion

Modeling Excitable Systems: Reentrant Tachycardia AJP 78, 56(2010) J.L. Lancaster, E.M. Leise, and E.H. Hellen or pdf. Describes electronic and mathematical models, used to model a cardiac arrhythmia and its treatment. Here is SPICE prediction for the 3-transistor circuit. Uses Ebers-Moll model of npn transistor.
fitzhugh-nagumo

Genetic Network Modeling:   Synthetic genetic networks can be constructed to perform desired functions. We explore potential dynamics of these networks. Hellen EH, Volkov E, Kurths J, Dana SK (2011) An Electronic Analog of Synthetic Genetic Networks. PLoS ONE 6(8): e23286. doi:10.1371/journal.pone.0023286 http://www.plosone.org/article/info%3Adoi%2F10.1371%2Fjournal.pone.0023286


α = 176, n = 4.1
β ratio = 0.45:0.1:0.1
k0 = 1, k1 = 0.01
η = 2, κ ≈ 10
Q varied

α = 176, n = 4.1
β ratio = 0.45:0.1:0.1
k0 = 1, k1 = 0.01
η = 2, κ ≈ 10
Q = 0.23

α ≈ 88, n ≈ 2.5+
β's = 0.1
k0 = 1, k1 = 0.025
η = 2, κ increases
Q = 0

α ≈ 88, n ≈ 2.5
β's = 0.1
k0 = 1, k1 = 0.025
η = 2, κ increases
Q = 0

Nonlinear Damping of the LC Circuit using Anti-parallel Diodes. Interesting amplitude dependent behavior occurs when anti-parallel diodes replace the resistor in a RLC circuit. The circuit becomes a nonlinearly damped harmonic oscillator obeying the equation:

  where 

More info. or get article Hellen and Lanctot, Am. J. Phys. 75, 326(2007) Or click for pdf

Logarithmic decay:   Diode-capacitor circuit.  What happens when the resistor in an RC decay circuit is replaced by a diode?  The decay changes from exponential to logarithmic.  (At least to a good approximation for 5 decades of time.) Click for more on Diode-Capacitor Decay. For reprint of article, Click here (pdf) or, this article may also be found at Edward H. Hellen, Am. J. Phys. 71, 797(2003).

Lorenz Chaotic Attractor: Jarrett Lancaster did this one night in the lab.

Padé-Laplace Curve-Fitting and Signal Averaging

Pade-Laplace is an intriguing method for finding the decay constants in a multi-exponential decay. Unlike standard methods of curve-fitting, it does not use the sum of the square of the errors and does not assume the number of decay constants. Instead it uses Laplace transforms, Pade approximants, Taylor series expansion, matrix inversion, and finding the zeroes of polynomials in order to determine the number of decay constants, their values, and their amplitudes. How it works

Here Padé-Laplace analysis is used on signal averaged data obtained from a simple circuit that produces multi-exponential voltage decays in the presence of noise. Click for article (pdf) or Edward H. Hellen, Am. J. Phys. 73, 871(2005).

Other Stuff

Here is Classic Beauty, perfection that will never be repeated.

Would you like a python-based MatLab-like system on your pc, for free? Here is how: Install MatLab-like

SPICE stuff. Ngspice is the free open source circuit simulator.

Virtualbox is nice and easy for creating virtual machines. {On (Fedora) linux, need to have installed the packages: gcc, dkms, kernel-devel, and make.}

Restringing wood tennis racquets.

LaTex Equation to Image This tool converts latex equation code into an image, like this one  .  {png better than gif}

Fun Electronics (No semiconductors here)  6L6,  Thordarson, Eico

De La Rive Electric Discharge Tube (fig at right) in action qv_x_B. To save the qv x B movie, right-click here and "Save Target as" using extension .wmv instead of .html.
De La Rive tube

Total Internal Reflection Fluorescence Microscopy TIRFM (avi of VPython animation). This one shows the evanescent illumination: TIRFM2. These movies demonstrate how TIR is achieved through the microscope objective. See publication of early use of this method to measure hormone (epidermal growth factor) binding kinetics at cell surface. (Part of my Ph.D. thesis)

Nearby hiking Pic1 Pic2 Pic3 Mov1

AAPT Summer 05 Salt Lake City Pic 1 Pic 2 Pic 3

Python notes here 

Intro Physics python demos here

Publications

UNCG Physics